Isospin particle systems on quaternionic projective spaces

TL;DR

This paper constructs an isospin particle system on quaternionic projective spaces with BPST-instanton, introduces a quaternionic oscillator potential, and demonstrates superintegrability with novel third-order constants of motion.

Contribution

It develops a new superintegrable system on quaternionic projective spaces with higher-order constants of motion, extending previous models on complex projective spaces.

Findings

System constructed on quaternionic projective spaces with BPST-instanton.

Addition of quaternionic oscillator potential yields superintegrability.

Presence of third-order constants of motion beyond quadratic ones.

Abstract

We construct the isospin particle system on $n$-dimensional quaternionic projective spaces in the presence of BPST-instanton by the reduction from the free particle on $(2n+1)$-dimensional complex projective space. Then we add to this system a "quaternionic oscillator potential" and show, that this oscillator-like system is superintegrable. We show, that besides the analogs of quadratic constants of motion of the spherical (Higgs) and $\mathbb{C}P^n $- oscillators, it possesses the third-order constants of motion, which are functionally independent from the quadratic ones.